Electronic Journal of Statistics

On a Gibbs sampler based random process in Bayesian nonparametrics

Stefano Favaro, Matteo Ruggiero, and Stephen G. Walker

Full-text: Open access

Abstract

We define and investigate a new class of measure-valued Markov chains by resorting to ideas formulated in Bayesian nonparametrics related to the Dirichlet process and the Gibbs sampler. Dependent random probability measures in this class are shown to be stationary and ergodic with respect to the law of a Dirichlet process and to converge in distribution to the neutral diffusion model.

Article information

Source
Electron. J. Statist., Volume 3 (2009), 1556-1566.

Dates
First available in Project Euclid: 4 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1262617419

Digital Object Identifier
doi:10.1214/09-EJS563

Mathematical Reviews number (MathSciNet)
MR2578838

Zentralblatt MATH identifier
1326.60105

Keywords
Random probability measure Dirichlet process Blackwell-MacQueen Pólya urn scheme Gibbs sampler Bayesian nonparametrics

Citation

Favaro, Stefano; Ruggiero, Matteo; Walker, Stephen G. On a Gibbs sampler based random process in Bayesian nonparametrics. Electron. J. Statist. 3 (2009), 1556--1566. doi:10.1214/09-EJS563. https://projecteuclid.org/euclid.ejs/1262617419


Export citation

References

  • [1] Antoniak, C.E., 1974. Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems., Ann. Statist. 2, 1152–1274.
  • [2] Billingsley, P., 1968., Convergence of probability measures. Wiley, New York.
  • [3] Blackwell, D., 1973. Discreteness of Ferguson selections., Ann. Statist. 1 356–358.
  • [4] Blackwell, D., MacQueen, J.B., 1973. Ferguson distributions via Pólya urn schemes., Ann. Statist. 1, 353–355.
  • [5] Dawson, D.A., 1993., Measure-valued Markov processes. Ecole d’Eté de Probabilités de Saint Flour XXI. Lecture Notes in Mathematics 1541. Springer, Berlin.
  • [6] Erhardsson, T., 2008. Nonparametric Bayesian inference for integrals with respect to an unknown finite measure., Scand. J. Statist. 35, 354–368.
  • [7] Ethier, S.N., Kurtz, T.G., 1986., Markov processes: characterization and convergence. Wiley, New York.
  • [8] Ethier, S.N., Kurtz, T.G., 1993. Fleming-Viot processes in population genetics., SIAM J. Control Optim. 31, 345–386.
  • [9] Favaro S., Walker S. G., 2008. A generalized constructive definition for Dirichlet processes., Statist. Probab. Lett. 78 2836–2838.
  • [10] Feigin, P. D., Tweedie, R. L., 1989. Linear functionals and Markov chains associated with Dirichlet process., Math. Proc. Cambridge Philos. Soc. 105 579–585.
  • [11] Ferguson, T.S., 1973. A Bayesian analysis of some nonparametric problems., Ann. Statist. 1, 209–230.
  • [12] Ferguson, T. S., 1974. Prior distributions on spaces of probability measures., Ann. Statist. 2, 615–629.
  • [13] Gelfand, A.E., Smith, A.F.M., 1990. Sampling-based approaches to calculating marginal densities., J. Amer. Statist. Assoc. 85, 398–409.
  • [14] Guglielmi, A. Tweedie, R. L., 2001. Markov chain Monte Carlo estimation of the law of the mean of a Dirichlet process., Bernoulli 7 573–592.
  • [15] Jarner, S. F., Tweedie, R. L., 2002. Convergence rates and moments of Markov chains associated with the mean of Dirichlet processes., Stochastic Process. Appl. 101 257–271.
  • [16] Lijoi, A. and Prünster, I., 2009. Distributional properties of means of random probability measures., Statist. Surveys 3, 47–95.
  • [17] Roberts, G.O., Rosenthal, J.S., 2004. General state space Markov chains and MCMC algorithms., Probab. Surv., 1, 20–71.
  • [18] Ruggiero, M. and Walker, S. G. (2009). Bayesian nonparametric construction of the Fleming-Viot process with fertility selection., Statist. Sinica, 19 707–720.
  • [19] Sethuraman, J., 1994. A constructive definition of Dirichlet priors., Statist. Sinica. 4 639–650.
  • [20] Walker, S. G., Hatjispyros, S. J. and Nicoleris, T. (2007). A Fleming-Viot process and Bayesian nonparametrics., Ann. Appl. Probab. 17, 67–80.
  • [21] Wilks, S., 1962., Mathematical Statistics. John Wiley, New York.